The gauge-theory tag has no usage guidance.

**14**

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**2**answers

502 views

### Are there results from gauge theory known or conjectured to distinguish smooth from PL manifolds?

My question begins with a caveat: I sometimes spend time with topologists, but do not consider myself to be one. In particular, my apologies for any errors in what I say below — corrections are ...

**1**

vote

**1**answer

903 views

### Wedge Product of Lie Algebra Valued One-Form

I've been reading about the formal structure of gauge theories and am a little confused by the notation. Could someone clarify this for me?
Suppose that $A$ is a Lie algebra valued 1-form ...

**2**

votes

**1**answer

204 views

### Isometric embedding of a compact Lie Group in $M(n,\mathbb{C})$

Greetings,
Let $G$ be a compact Lie group with a bi-invariant inner product $h$ on it. Can one embedd $G$ in $M(n,\mathbb{C})$ isometrically for some $n \in \mathbb{N}$. By isometrically I mean that ...

**6**

votes

**1**answer

355 views

### Kähler form on complex Lie group

Hallo,
Let $G$ be a semi-simple, compact Lie Group. Consider its complexification $G_{\mathbb{C}}$. Does there exist a Kähler structure on $G_{\mathbb{C}}$ which is $G$-invariant (maybe in a ...

**5**

votes

**1**answer

775 views

### Relation of SW and Donaldson Invariant

My question is:
I am request for the reference that Is there any relationship between the Seiberg-Witten Invariant and Donaldson's Invariant? Or the relationship between Seiberg-Witten Moduli Space ...

**5**

votes

**0**answers

248 views

### Seiberg-Witten curve for product SU(2)^N gauge theories

In equation 2.10 of this article, the author gives the Seiberg-Witten curve for a $U(N)$ gauge theory with $L<2N$ massive flavours with masses given by $m_i$ as:
$y^{2}=\left\langle \mathrm{det}\...

**6**

votes

**2**answers

848 views

### Floer homology and Invariants for Einstein Field Equations?

Motivation: There have been the instanton (anti-self dual connection) solutions to the Yang-Mills equation $d_A^\ast F_A=0$ which extremize the YM energy $\int_M|F_A|^2$, leading to the Donaldson ...

**3**

votes

**1**answer

473 views

### The number of simply connected 4-dimension manifold

For a simply connected four-dimension manifold, we know the Freedmen's work.
My question is: For every integer N, Is the number of simply connected 4-manifolds which the second betti number is ...

**10**

votes

**0**answers

292 views

### What is known about the Yang-Mills stratification over 3-manifolds?

Rade proved in his thesis (Crelle's Journal, 1992, available here: digizeitschriften.de/dms/toc/?PPN=PPN243919689) that if $E\rightarrow M$ is a $U(n)$-bundle over a 3-manifold, then the gradient flow ...

**7**

votes

**5**answers

797 views

### Symplectic structures from Lagrangians?

In Witten's paper 'Quantization of Chern-Simons Gauge Theory with Complex Gauge Group,' he makes the statement (p. 35) that the symplectic form on the moduli space of flat $G_\mathbb{C}$ connections ...

**5**

votes

**1**answer

354 views

### How to compute the Monopole Floer Homology for Surface $\times S^1$ ?

We know that Monopole Floer homology of a 3-manifold $M$ depends on a spin-c structure. My question is that if $M$ is $F\times S^1$ ($F$ is a surface of genus larger than 1) then how can we compute ...

**11**

votes

**5**answers

1k views

### Yang-Mills and Chern-Simons functionals as Morse functions

Can the Yang-Mills or Chern-Simons action functionals be considered as [possibly perfect] Morse functions? I assume we would be in an equivariant scenario due to considering the configuration spaces ...

**0**

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**0**answers

214 views

### Variation of the Chern connection according to the variation of hermitian metric

Whats is the relation between the Chern connections of tow Hermitian metrics in a holomorphic vector bundle?

**1**

vote

**1**answer

347 views

### Reference request: Seminal papers in gauge-theoretic mathematics [closed]

Following on from previous question I was also searching for seminal papers in gauge theory.
Would be greatly appreciative of references to such.

**5**

votes

**2**answers

542 views

### Reference request: Introductions to current mathematics derived from / related to gauge theories

I was searching for introductions to current mathematics related to gauge theories.
Can someone suggest some good references?
E.g.
Topics in Physical Mathematics by K. Marathe

**6**

votes

**0**answers

392 views

### Has anyone seen this Hitchin-like system?

Let $(M,g)$ be a riemannian manifold and let $P\to M$ be a principal $G$-bundle with connection $A$. Let $\alpha \in \Omega^1(M;\mathrm{ad}P)$ be a one-form on $M$ with values in the adjoint bundle $\...

**6**

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**3**answers

2k views

### Is the Lie algebra-valued curvature two-form on a principal bundle P the curvature of a vector bundle over P?

I am an analyst struggling through some geometry used in physics.
Some background: For some Lie group $G$, let $P$ be a principal $G$-bundle over the smooth manifold $M$. Let $\omega$ be a connection ...

**7**

votes

**1**answer

1k views

### K.Uhlenbeck's preprint “A priori estimates for Yang-Mills fields”

Does anyone have a copy of the unpublished preprint of Karen Uhlenbeck A priori estimates for Yang-Mills fields from around 1986?
It appears to have circulated for some time, and it is quoted in ...

**3**

votes

**2**answers

979 views

### Why must a reducible flat SU(2)-connection over a homology sphere be trivial?

Let $M$ be a homology sphere. Suppose $P=M\times SU(2)$ is the trivial $SU(2)$ principal bundle. Let $R$ be all reducible connections on $P$. Here $A$ in $R$ is reducible if the gauge transformation ...

**2**

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**0**answers

252 views

### Question in complex analysis arising from large $N$ gauge theory

This is a question in complex analysis that comes up in the treatment of large
$N$ gauge theory with gauge group $SU(N)$ in the 't Hooft limit where $N$ is taken to infinity with $\lambda=g^2 N$ fixed ...

**24**

votes

**4**answers

2k views

### What is the precise statement of the correspondence between stable Higgs bundles on a Riemann surface, solutions to Hitchin's self-duality equations on the Riemann surface, and representations of the fundamental group of the Riemann surface?

I am trying to find the precise statement of the correspondence between stable Higgs bundles on a Riemann surface $\Sigma$, (irreducible) solutions to Hitchin's self-duality equations on $\Sigma$, and ...

**24**

votes

**5**answers

3k views

### Is symplectic reduction interesting from a physical point of view?

Do you think that symplectic reduction (Marsden Weinstein reduction) is interesting from a physical point of view? If so, why? Does it give you some new physical insights?
There are some possible ...

**1**

vote

**2**answers

357 views

### module of sections of the horizontal bundle

Some times ago I posted this question here. There I carelessly assumed that if you have a set of sections of a vector bundle which span every fiber pointwise, they also generate the module of smooth ...

**4**

votes

**1**answer

409 views

### How do you exponentiate a section of the adjoint bundle to get a gauge transformation?

Suppose $E$ is a vector bundle with structure group $G$ and let $P = F(E)$ be the frame bundle. Let $\mathfrak{g}_P$ denote the associated bundle to the adjoint representation of $G$ on its Lie ...

**6**

votes

**3**answers

582 views

### Literature for gauge field theory on the lattice in geometrical formulation

I have found an article by Huebschmann, Rudolph and Schmidt: http://www.springerlink.com/content/b8v216v0m8h16264/ about "A Gauge Model for Quantum Mechanics on a Stratified Space" and I am very ...

**1**

vote

**1**answer

332 views

### Reference for some elementary facts about principal bundles

Let $(P,\pi,B,G)$ be a principal bundle with total space $P$, base $B$, projection $\pi$ and structure group $G$.
Now I am searching for a good reference (with proofs) for the following facts:
1) ...

**7**

votes

**0**answers

563 views

### Central Yang-Mills connections, and flat connections with prescribed holonomy

Let $X$ be $\Sigma^g$ which is the Riemann surface of genus $g$, and consider a trivial $G$-bundle over it.
1) In this $2$-d setting, the space of Yang-Mills central connections is the set of ...

**2**

votes

**1**answer

418 views

### gauge theory construction of vector bundles on singular varieties

This is sort of a follow-up to:
Gauge theory construction of moduli of vector bundles
If I have a complex compact algebraic curve with at worst nodal singularities, is there an analytic description ...

**6**

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**3**answers

2k views

### Gauge theory construction of moduli of vector bundles

Given a smooth projective complex variety $X$, instead of using Mumford's GIT to construct the moduli of rank $n$ topologically trivial vector bundles, we can also take the gauge theory approach.
To ...

**1**

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**3**answers

544 views

### Can all G-connections on a Riemann surface X be induced by maps from X to G

There is the invariant Maurer–Cartan 1-form on a compact semi-simple Lie group G. So if we pull it back using a map from X to G then we get a G-connection on X. The question is, can all G-connections ...

**9**

votes

**1**answer

987 views

### Seiberg-Witten theory on 4-manifolds with boundary

What generalizations of Seiberg-Witten theory to 4-manifolds with boundary do exist?
I would be especially interested in theories which "behave good" under gluing along the boundary (comparable to ...

**3**

votes

**2**answers

663 views

### Deriving symmetries of a Gauge theory

Hello,
I don't know if this is a good place for exposing my problem but I'll try...
I have a gauge theory with action:
$S=\int\;dt L=\int d^4 x \;\epsilon^{\mu\nu\rho\sigma} B_{\mu\nu\;IJ} F_{\mu\nu}...

**9**

votes

**3**answers

1k views

### Looking for reference on gauge fields as connections.

Can anyone give me references where I would see a detailed exposition of how to translate gauge field theory as known to physicists into the language of connections. I am looking for a detailed ...

**7**

votes

**1**answer

751 views

### Casson's invariant and the trivial connection contribution to witten's 3-manifold invariant

This question might turn out not to make any sense, but here it is: Witten's (and Reshetikhin and Turaev's) 3-manifold invariant can be "defined" as an integral over the space of connections on the ...

**5**

votes

**1**answer

501 views

### Monopole classes on hyperbolic 3-manifolds

Let $M$ be a closed hyperbolic $3$-manifold, and $e \in H^2(M)$ an integral cohomology class which is the first Chern class of a $Spin^c$ structure on $M$. Suppose there is a solution to the monopole ...

**12**

votes

**3**answers

2k views

### The “miracle” of Heegard Floer.

Taking tori in symmetric products and "miraculously" proving that the Floer homology is independent of choices always seemed, well, miraculous. Some time ago Max Lipyanski explained to me the origins ...